Solution
Level Three in a Three-Player Beauty Contest
Show the problem again
Three players each name a real number in [0, 100]. The target is 2/3 of the average of all three numbers, including your own, and the player closest to the target wins. A level-0 player names 50. For k ≥ 1, a level-k player believes both opponents are level-(k−1) and names the number that equals the target exactly (such a number beats both opponents outright, since they name the same number). What number does a level-3 player name? Give an exact fraction.
Worked solution
The answer is 3200/343, about 9.33. Suppose both opponents name x and you name y. Hitting the target means y = (2/3)·(2x + y)/3, because your own number is part of the average. Then 9y = 4x + 2y, so y = 4x/7. The factor 4/7 is smaller than the naive 2/3 because your own number is part of the average. Starting from 50, level 1 names 200/7, level 2 names 800/49, and level 3 names 3200/343. In the limit the chain converges to 0, which is the unique Nash equilibrium.
Source: Beauty-contest game after Nagel (1995) and Keynes; level-k reasoning after Stahl and Wilson, Nagel. Three-player variant with own guess included written for AxiomIQ.